Colin Fox

Richard A. Norton, Colin Fox, Malcolm E. Morrison

We develop a finite-dimensional approximation of the Frobenius-Perron operator using the finite volume method applied to the continuity equation for the evolution of probability. A Courant-Friedrichs-Lewy condition ensures that the approximation satisfies the Markov property, while existing convergence theory for the finite volume method guarantees convergence of the discrete operator to the continuous operator as mesh size tends to zero. Properties of the approximation are demonstrated in a computed example of sequential inference for the state of a low-dimensional mechanical system when observations give rise to multi-modal distributions.

J. Andrés Christen, Colin Fox, Diego Andrés Pérez-Ruiz et al.

Generalized Gibbs kernels are those that may take any direction not necessarily bounded to each axis along the parameters of the objective function. We study how to optimally choose such directions in a Directional, random scan, Gibbs sampler setting. The optimal direction is chosen by minimizing to the mutual information (Kullback-Leibler divergence) of two steps of the MCMC for a truncated Normal objective function. The result is generalized to be used when a Multivariate Normal (local) approximation is available for the objective function. Three Gibbs direction distributions are tested in highly skewed non-normal objective functions.